Proof of the Normal Sum Theorem

  The reason that I haven't been working on posts lately is that I've been doing random walk simulations in order to determine the relative contribution to the global temperature anomaly. A good introduction to the subject of random walks is Don S. Lemons, An Introduction to Stochastic Processes in Physics. In the book Lemons discusses probability distributions, their properties and gives the normal sum theorem which states that a variable z = x + y which is the sum of two variables with normal distributions also has a normal distribution. His justification for the theorem is that the mean and variance for z is the sum of the means and variances for the two distributions. This lacks some rigor since he doesn't prove that the resulting distribution is actually a normal one. It is not very difficult to do so.

  To simplify the proof we set the mean of the two distributions to zero and let x and y vary independently over their possible values. The density for joint distribution, P(x,y), is defined for an element of area with sides dx and dy and is the product of the two individual normal probability densities. Instead of using combinations of x and y it is easier to use z and w = x - y as our variables and express P(x,y)dxdy in terms of them. The directions for the axes of z and w are perpendicular to each other but at angles to x and y so we find that the element of area has a factor of 1/2 associated with it.

  We next replace x and y in the exponential with their formulas in terms of z and w and simply the result defining the new constants λ, μ and α .

  To get the probability distribution for z we integrate the joint distribution over all values of w. We can simply the integral by factoring out the z² term in the exponential leaving an integral, I(λ,α,z), which can be further simplified by completing the square and evaluated using the standard formula for the Gaussian integral.

  It is fairly easy to show that φ(z) is a normal distribution with variance σ_eff² = σ² + s².